Book•
Riemannian Geometry of Contact and Symplectic Manifolds
08 Jan 2002-
TL;DR: In this article, the authors describe a complex geometry model of Symplectic Manifolds with principal S1-bundles and Tangent Sphere Bundles, as well as a negative Xi-sectional Curvature.
Abstract: Preface * 1. Symplectic Manifolds * 2. Principal S1-bundles * 3. Contact Manifolds * 4. Associated Metrics * 5. Integral Submanifolds and Contact Transformations * 6. Sasakian and Cosymplectic Manifolds * 7. Curvature of Contact Metric Manifolds * 8. Submanifolds of Kahler and Sasakian Manifolds * 9. Tangent Bundles and Tangent Sphere Bundles * 10. Curvature Functionals and Spaces of Associated Metrics * 11. Negative Xi-sectional Curvature * 12. Complex Contact Manifolds * 13. Additional Topics in Complex Geometry * 14. 3-Sasakian Manifolds * Bibliography * Subject Index * Author Index
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TL;DR: In this paper, a twistorial interpretation of geometric structures on a Riemannian manifold, as sections of homogeneous fibre bundles, was given, following an original insight by Wood (2003).
Abstract: We give a twistorial interpretation of geometric structures on a Riemannian manifold, as sections of homogeneous fibre bundles, following an original insight by Wood (2003). The natural Dirichlet energy induces an abstract harmonicity condition, which gives rise to a geometric gradient flow. We establish a number of analytic properties for this flow, such as uniqueness, smoothness, short-time existence, and some sufficient conditions for long-time existence. This description potentially subsumes a large class of geometric PDE problems from different contexts.
As applications, we recover and unify a number of results in the literature: for the isometric flow of ${\rm G}_2$-structures, by Grigorian (2017, 2019), Bagaglini (2019), and Dwivedi-Gianniotis-Karigiannis (2019); and for harmonic almost complex structures, by He (2019) and He-Li (2019). Our theory also establishes original properties regarding harmonic flows of parallelisms and almost contact structures.
19 citations
TL;DR: In this article, point-wise slant submanifolds of almost contact and almost contact 3-structure manifold are introduced and characterized, and necessary and sufficient conditions for a pointwise SLA of a 3-Sasakian manifold to be a SLA are given.
Abstract: In this paper, we introduce point-wise slant submanifolds of almost contact and almost contact 3-structure manifolds. We characterize them, give some examples, and obtain necessary and sufficient conditions for a point-wise slant submanifold of a 3-Sasakian manifold to be a slant submanifold. Moreover, we show that there exist no proper Sasakian point-wise 3-slant submanifolds.
19 citations
Additional excerpts
...(1) and (2) imply φξ = 0 and ηoφ = 0 [2]....
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TL;DR: In this article, the authors considered two magnetic fields on the 3-torus obtained from two different contact forms on the Euclidean 3-space and studied when their corresponding normal magnetic curves are closed.
Abstract: We consider two magnetic fields on the 3-torus obtained from two different contact forms on the Euclidean 3-space and we study when their corresponding normal magnetic curves are closed. We obtain periodicity conditions analogues to those for the closed geodesics on the torus.
19 citations
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TL;DR: In this paper, the first introduction to geometric structures on $TM\oplus T^*M$ was provided, which contains definitions and characteristic properties of generalized complex, generalized Kaehler, generalized (normal, almost) contact and generalized Sasakian structures.
Abstract: This is an expository paper, which provides a first introduction to geometric structures on $TM\oplus T^*M$ The paper contains definitions and characteristic properties of generalized complex, generalized Kaehler, generalized (normal, almost) contact and generalized Sasakian structures A few of these properties are new
19 citations
TL;DR: In this paper, the triviality of the (0, 2)-component of the basic Euler class characterizes the existence of compatible Sasakian metrics for given small deformations of the Reeb flow as transversely holomorphic Riemannian flows.
Abstract: Deformations of the Reeb flow of a Sasakian manifold as transversely K\"ahler flows may not admit compatible Sasakian metrics anymore. We show that the triviality of the (0,2)-component of the basic Euler class characterizes the existence of compatible Sasakian metrics for given small deformations of the Reeb flow as transversely holomorphic Riemannian flows. We also prove a Kodaira-Akizuki-Nakano type vanishing theorem for basic Dolbeault cohomology of homologically orientable transversely K\"ahler foliations. As a consequence of these results, we show that any small deformations of the Reeb flow of a positive Sasakian manifold admit compatible Sasakian metrics.
19 citations
Cites background from "Riemannian Geometry of Contact and ..."
...By the last sentence of Section 6.4 of Blair [3] or Proposition 6.5.14 of Boyer and Galicki [5], Lemma 2.8....
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...Basic examples of Sasakian manifolds are circle bundles associated to positive holomorphic line bundles over Kähler manifolds, links of isolated singularities of complex hypersurfaces defined by weighted homogeneous polynomials and contact toric manifolds of Reeb type (see Boyer and Galicki [4] and Blair [3])....
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...Basic examples of Sasakian manifolds are circle bundles associated to positive holomorphic line bundles over Kähler manifolds, links of isolated singularities of complex hypersurfaces defined by weighted homogeneous polynomials and contact toric manifolds of Reeb type (see Boyer and Galicki [4] and Blair [3])....
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